Sādhanā Vol. 33, Part 3, June 2008, pp. 299–327. © Printed in India
Axisymmetric fretting analysis in coated cylinder
M RAMESH∗ , SATISH V KAILAS and K R Y SIMHA
Department of Mechanical Engineering, Indian Institute of Science,
Bangalore 560 012
e-mail: mramesh@mecheng.iisc.ernet.in
Abstract. Fretting is essentially a contact fatigue phenomenon, although bulk
stresses and material properties contribute to final failure. The near surface state of
stress developed under oscillatory contact between machine elements plays a major
role in deciding the severity of fretting. It is possible to enhance tribological properties by coating the surface. There is rather scanty literature available on fretting
analysis of coated components. Presence of such coatings has a large influence on
the near surface state of stress. The effect of coatings on the severity of fretting is
the focus of this paper. Results obtained for both hard and soft coatings are compared with the results obtained for the homogeneous case. The component geometry and loading are chosen to be cylindrical to enable 3D elastic axisymmetric
fretting analysis. The results are compared with 2D models (strip and half-plane) to
examine their utility and validity for understanding axisymmetric fretting. Contact
pressure and frictional shear loading cases are solved separately and superposed
appropriately depending on the coefficient of friction considered. Results for different values of coefficient of friction and elastic mismatch are illustrated through
contour plots of stresses and strains. These results are expected to be helpful for
identifying fretting failure zones and fracture mechanisms in coated components.
Analytical results presented here could serve as useful benchmarks for calibrating
numerical codes and experimental techniques.
Keywords.
Coated cylinder; axisymmetric analysis; static; Fourier transform.
1. Introduction
Fretting is essentially a contact fatigue phenomenon wherein the failure is predominantly
localised near the surface (Farris et al 2003). This is indeed the case at least during the initial
stages until the bulk stresses dominate and fatigue damage penetrates into the bulk material.
The nature of contact between engineering components determines the near surface state of
stress, which in turn controls the severity of fretting. There are cases where the engineering
components are made of materials with good mechanical properties but poor tribological
properties, for example, Al alloys used in aerospace industries. It is common to coat the
substrate, or provide surface treatment. The coating material is chosen to give good tribological
∗ For
correspondence
299
300
M Ramesh, Satish V Kailas and K R Y Simha
properties. Such coatings exert a larger influence on the near surface state of stress. The effect
of such coatings and their mechanical properties on the near surface state of stress is the focus
of this paper.
There is a significant effort by metallurgists and materials scientists to synthesize new
coating materials and techniques to combat corrosion and heat loss. Unfortunately, coatings
are unreliable under fatigue loading in general and fretting in particular due to interfacial
defects. Coatings in general can be of two types: hard and soft. Soft coatings as solid lubricants reduce the coefficient of friction though under fretting the coatings get worn out. This
brings the substrate to come into contact with the pad which will lead to crack initiation.
Thus, the soft coating delays crack initiation but does not prevent crack propagation. Shot
peening treatment before the application of coating is found to retard crack propagation
rate. However, care should be taken not to introduce cracks during the shot peening treatment. Hard coatings in general are brittle. They protect the substrate from subsequent wear
and crack initiation, but they fail by delamination which in turn may even reduce fretting
fatigue life. The main reason being the weak interface. Golden et al (2007); Rajasekaran
et al (2008); Xu et al (2007) discussed these issues along with effects of residual stress and
porosity.
Even in the absence of any coating, formation of transfer/interfacial layer between contacting parts can influence the state of stress. The coated cylinder model can also shed new
light on the tribology of solid lubricants, transfer/interfacial layer, as discussed by Williams
(2005). However, in the context of fretting where the amplitudes of relative motion between
the contacting bodies are generally small, Hager et al (2004) reported the formation of hard
tribologically transformed surface (TTS) in the mixed fretting regime where both fretting and
wear are simultaneously present.
In the case of coated components fretting cracks initiated at the surface combine forces
with cracks initiated at the interface to drastically reduce the fretting life. The stresses along
the interface are critical to assessing the reliability of fretting components. The interfacial
shear stress and the stress normal to the interface (particularly when it is tensile) control the
fatigue life expectancy of coatings. Depending on the elastic mismatch between the coating
and the substrate, severe stress gradients could be generated in the film.
If the coating bonds strongly with the substrate then the coating delays crack initiation and
retards crack propagation as shown by experiments. Nishida et al (2003) showed experimentally that presence of lubricating anodic film delays crack initiation and also resulting in low
crack growth rates. Okane et al (2003) showed that WC-Co spray over NiCrMo steel retards
fretting fatigue crack initiation and lowers propagation rates. Thus, fretting fatigue strength
of a component can be enhanced by coatings. Analysis of coatings in fretting fatigue theory
is not emphasised in the literature. This paper is probably the first attempt aimed at a comprehensive stress analysis of coated cylinders under fretting loading.
In the context of axisymmetric cylindrical interfacial cracks Erdogan & Ozbek (1969);
Ozbek & Erdogan (1969) followed by Farris et al (1989) and Itou (1990) analysed cylindrical
interfacial cracks in infinite domain. Engineering components require solutions inside finite
domain such as coated cylinders and shafts. In this class of problems before we proceed with
fracture mechanics analysis, it is necessary to perform the stress analysis when the cracks
are absent. This is the primary motivation for this paper. Here we show that fretting loads
generate both interfacial shear and tensile stress in the coating.
Analysis of cylindrical geometries like shafts and shrink-fit assemblies generally require
the application of Fourier transform technique. A comprehensive introduction to fretting analysis for a shaft with press fitted hub using Fourier transform was given by
Axisymmetric fretting analysis in coated cylinder
301
Dobromirsky & Smith (1986). They simulated the effect of axial cyclic loading by superposing normal and shear stress on the cylindrical surface. They claimed that the pressure distribution at the hub shaft interface will become uniform after a few cycles due
to wear of edges by fretting which otherwise would create peak stresses at the edges.
They assumed as a first approximation zero shear stress in the stick region and uniform
shear stress in the slip region whose magnitude depends on the coefficient of friction and
the applied pressure. The above assumptions are considered valid and the same is followed in this work. Further, the contact length 2a, stick zone length 2c, contact pressure
p0 and frictional shear q0 magnitude at the surface are all interdependent in the actual
problem. However, for simplicity reasonable values of the parameters are chosen independently in this paper and the effect of each parameter on the near surface state of stress is
discussed.
There is a common tendency to interpret fretting fatigue crack initiation based on indentation stress field (ISF). In reality indentation stress fields require the solution to mixed boundary
problems including the possibility of stress and strain singularities at the edges. Singularities,
if present in fretting, are neutralised by repeated relative motion between contacting parts.
Fretting in flat and round configuration (Ciavarella & Demelio 2001) in the context of turbine
blade/disk interface present no edge singularities. The portion which sticks is mainly under
Hertzian pressure loading and the shear stress effect is confined to the ends experiencing slip.
The stick/slip front moves inward from the ends as the shear load builds up. Eventually, at
a sufficiently high shear load, gross sliding occurs. In fretting analysis, however, we focus
our attention on extremely small amplitudes of slip (< 50μ m) at the ends of the punch. It is
assumed that the stick/slip boundary is fixed. With this idealised modelling, it is convenient
to perform the stress analysis for a uniform pressure and shear profile. Thus, we reexamine
and extend the work of Dobromirsky & Smith (1986) for coated cylinders which brings in
the additional complexity due to elastic mismatch.
In addition, we examine the utility of 2D plane strain results pertaining to a coated strip and
a half-plane under identical loading. This will help us interpreting 2D fretting experiments
for coated cylinders. Experimental research on fretting phenomena is generally investigated
using specimens in the form of strips and plates (ASTM STP 1425). There is therefore a
need to compare and correlate stresses generated by fretting loading in these three analogous
geometries (cylinder, strip and half-plane, figures 1, 2 and 3) to examine the utility of simplified
strip and plate analysis. Comparison of stresses in strip and half-plane has already been done
by Fellows et al (1995) for the homogeneous case which we now extend to the case of coated
cylinder, strip and half-plane.
The coated cylinder configuration is shown in figure 1. The configuration for coated strip
and coated half-plane are shown in figures 2 and 3. The results are obtained for the following
cases:
•
•
•
•
•
different ratios of elastic shear moduli of coating and shaft (μA /μB )
different values of coefficient of friction (φ)
different geometries: cylinder, strip and half-plane
different ratios of a/R1
different ratios of c/a.
The actual fretting problem entails both pressure and shear loading which are analysed
separately and superposed appropriately to simulate fretting loading. The tractions applied
by Dobromirsky & Smith (1986) comprised of uniform pressure and shear (figure 4). The
same idea is followed in this paper for the case of coated cylinder subjected to fretting.
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M Ramesh, Satish V Kailas and K R Y Simha
Figure 1. Coated cylinder with cylindrical co-ordinate system.
Figure 2. Coated strip with Cartesian coordinate system.
The uniform pressure/shear profile assumed in this paper provide a fundamental basis for
constructing other profiles. For example, Hertzian and triangular profiles represent reality
better. However, the corresponding computational complexity increases especially for coated
cylinder. We therefore consider the homogeneous case separately. Another issue of some
practical significance concerns the roughness of surfaces under fretting contact. Machined
surfaces often exhibit regular features during the initial stages of fretting such as sinusoidal
variation. The analysis for sinusoidal corrugation, for example, is possible. It is also important
to bear in mind the intrinsically dynamic nature of fatigue in general and fretting in particular.
This issue assumes significance when the fretting frequency lies close to the natural frequency
Figure 3. Coated half-plane with Cartesian
co-ordinate system.
Axisymmetric fretting analysis in coated cylinder
303
Figure 4. Traction profiles.
of the components under study. This aspect requires an advance analysis using elastodynamic
theory.
2. Formulation
2.1 Infinitely long coated cylinder
Axisymmetric formulation based on Love’s stress function satisfies the biharmonic equation
∇ 2 ∇ 2 = 0 (Timoshenko & Goodier 1970). The method of Fourier integral transform is
used to obtain the solution for the stress function, the stresses and the displacements. For
more details the reader is referred to equations 1 to 16 of our paper entitled ‘Near surface
stress analysis strategies for Axisymetric fretting’ in this issue.
In the Fourier space the biharmonic equation (∇ 2 ∇ 2 = 0) becomes
2
2
d
d 1 d
1 d
2
2
+
+
(1)
−ξ
− ξ = 0.
dr 2
r dr
dr 2
r dr
The general solution to the biharmonic equation in the Fourier space is
I0 (ξ r)
K0 (ξ r)
1
I1 (ξ r)
K1 (ξ r)
+ A2
+ A3 ξ r
+ A4 ξ r
A = 3 A1
ξ
I0 (ξ R1 )
K0 (ξ R1 )
I0 (ξ R1 )
K0 (ξ R1 )
1
I1 (ξ r)
I0 (ξ r)
B = 3 B1
+ B3 ξ r
,
ξ
I0 (ξ R1 )
I0 (ξ R1 )
(2)
where Aj (j = 1, 2, 3, 4) and Bj (j = 1, 3) are functions of ξ to be determined from boundary
conditions. I0 , I1 , K0 and K1 are modified Bessel functions of first and second kind. Using
¯ A and ¯ B the stresses and displacements in the Fourier space for Cylinder
this solution for A (coating) and Cylinder B (core) are:
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M Ramesh, Satish V Kailas and K R Y Simha
Cylinder A (coating):
2μA urA = A1 (ξ )a11 (r) + A2 (ξ )a12 (r) + A3 (ξ )a13 (r) + A4 (ξ )a14 (r)
2μA uzA = A1 (ξ )a21 (r) + A2 (ξ )a22 (r) + A3 (ξ )a23 (r) + A4 (ξ )a24 (r)
σ rA = A1 (ξ )a31 (r) + A2 (ξ )a32 (r) + A3 (ξ )a33 (r) + A4 (ξ )a34 (r)
τ rzA = A1 (ξ )a41 (r) + A2 (ξ )a42 (r) + A3 (ξ )a43 (r) + A4 (ξ )a44 (r)
σ θA = A1 (ξ )a51 (r) + A2 (ξ )a52 (r) + A3 (ξ )a53 (r) + A4 (ξ )a54 (r)
σ zA = A1 (ξ )a61 (r) + A2 (ξ )a62 (r) + A3 (ξ )a63 (r) + A4 (ξ )a64 (r).
(3)
Cylinder B (core):
2μB urB = B1 (ξ )b11 (r) + B3 (ξ )b13 (r)
2μB uzB = B1 (ξ )b21 (r) + B3 (ξ )b23 (r)
σ rB = B1 (ξ )b31 (r) + B3 (ξ )b33 (r)
τ rzB = B1 (ξ )b41 (r) + B3 (ξ )b43 (r)
σ θB = B1 (ξ )b51 (r) + B3 (ξ )b53 (r)
σ zB = B1 (ξ )b61 (r) + B3 (ξ )b63 (r).
(4)
The expressions for aij and bij are given in Appendix A.1. The solution in real space can
be obtained by taking the inverse Fourier transform of the above expressions. The boundary
conditions for the problem at the outer radius r = R1 are the prescribed tractions; and, at the
interface radius r = R2 stress and displacement continuity
Outer boundary: σrA = P (z) and τrzA = Q(z)
Interface: σrA = σrB ; τrzA = τrzB ; urA = urB ; uzA = uzB .
The above conditions in the Fourier space are given below.
Outer boundary: σ rA = P ; τ rzA = Q
Interface: σ rA − σ rB = 0; τ rzA − τ rzB = 0; urA − urB = 0; uzA − uzB = 0.
The expressions for P and Q for uniform pressure and uniform double shear as in figure 4
are derived using auxiliary variable η.
Uniform pressure profile: P (η) = p0 for −a ≤ η ≤ a
∞
a
sin(ξ a)
P (ξ ) =
P (η) cos(ξ η)dz =
p0 cos(ξ η)dz = p0
.
(5)
ξ
0
0
Uniform shear profile: Q(η) = q0 for c ≤ η ≤ a
∞
a
cos(ξ c) − cos(ξ a)
Q(ξ ) =
Q(η) sin(ξ η)dz =
q0 sin(ξ z)dz = q0
.
ξ
0
c
(6)
Axisymmetric fretting analysis in coated cylinder
305
Also, p0 and q0 are related through the friction coefficient φ by q0 = φp0 . The auxiliary
variable η represents the coordinate z in the axisymmetric case and x in 2D plane Cartesian
coordinate system.
Substituting the expressions for the transformed stresses and displacements from equations 3 and 4 in the above mentioned boundary conditions we get the following equation in
the matrix form
[M]6×6 [A]6×1 = [P]6×1
where,
⎡
a31 (R1 ) a32 (R1 ) a33 (R1 ) a34 (R1 )
⎢
⎢ a41 (R1 )
⎢
⎢
⎢ a31 (R2 )
⎢
M=⎢
⎢ a41 (R2 )
⎢
⎢
⎢ a11 (R2 )
⎢
⎣
a21 (R2 )
⎡
(7)
A1
⎢
⎢ A2
⎢
⎢
⎢ A3
⎢
A=⎢
⎢ A4
⎢
⎢
⎢ B1
⎣
B3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥;
⎥
⎥
⎥
⎥
⎦
a42 (R1 ) a43 (R1 ) a44 (R1 )
a32 (R2 ) a33 (R2 ) a34 (R2 )
a42 (R2 ) a43 (R2 ) a44 (R2 )
a12 (R2 ) a13 (R2 ) a14 (R2 )
a22 (R2 ) a23 (R2 ) a24 (R2 )
⎡
P
0
0
0
0
⎤
⎥
⎥
⎥
⎥
−b31 (R2 )
−b33 (R2 ) ⎥
⎥
⎥
−b41 (R2 )
−b43 (R2 ) ⎥
⎥
⎥
μA
μA
− μB b11 (R2 ) − μB b13 (R2 ) ⎥
⎥
⎦
μA
μA
− μB b21 (R2 ) − μB b23 (R2 )
⎤
⎢ ⎥
⎢Q⎥
⎢ ⎥
⎢ ⎥
⎢ 0 ⎥
⎢ ⎥
P = ⎢ ⎥.
⎢ 0 ⎥
⎢ ⎥
⎢ ⎥
⎢ 0 ⎥
⎣ ⎦
0
The problems of pressure (Q = 0) and shear loading (P = 0) are solved separately.
The results are superposed appropriately to obtain the full solution. The six unknowns
A1 (ξ ), A2 (ξ ), A3 (ξ ), A4 (ξ ), B1 (ξ ) and B3 (ξ ) are determined by solving the above matrix
equation. Substituting these functions and inverting equations 3 and 4, the stresses and displacements are obtained. The formulation discussed in this paper corresponds to coated solid
cylinders. Since hollow cylinders are often used in applications the formulation can be easily
extended to a coated hollow cylinder by taking
I0 (ξ r)
K0 (ξ r)
1
I1 (ξ r)
K1 (ξ r)
B = 3 B1
+ B2
+ B3 ξ r
+ B4 ξ r
(8)
ξ
I0 (ξ R1 )
K0 (ξ R1 )
I0 (ξ R1 )
K0 (ξ R1 )
with two more extra boundary conditions along the inner boundary of the cylinder.
Owing to the axisymmetric nature, the tangential stress σθ becomes a principal stress. The
other two principal stresses are
σr + σz
σr − σz 2
2
.
(9)
+ τrz
±
2
2
Algebraically sorting these three values gives σ1 , σ2 and σ3 from which τoct can be obtained.
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M Ramesh, Satish V Kailas and K R Y Simha
2.2 Coated strip and half-plane
The formulation for strip and half-plane is done using plane Airy’s stress function in conjunction with Fourier transforms. The procedure followed is same as that of cylindrical case.
It should be noted that Fourier cosine transforms are required for symmetric loading. The
main results are listed below. The Airy stress function in the Fourier space for coated strip
and coated half-plane are as follows.
The Airy stress functions for the coating A and the strip B (figure 2) are:
A =
1
[A1 e−ξ(y−Y1 ) + A2 eξ(y−Y1 ) + A3 ξye−ξ(y−Y1 ) + A4 ξyeξ(y−Y1 ) ]
ξ2
B =
1
[B1 cosh(ξy)/e−ξ Y1 + B3 ξy sinh(ξy)/e−ξ Y1 ].
ξ2
(10)
The Airy stress functions for the coating A and the half-plane B (figure 3) are
A =
1
[A1 e−ξy + A2 eξy + A3 ξye−ξy + A4 ξyeξy ]
ξ2
B =
1
[B1 e−ξy + B3 ξye−ξy ].
ξ2
(11)
The stresses and displacements in the Fourier space are:
Coating A:
2μA uyA = A1 (ξ )a11 (y) + A2 (ξ )a12 (y) + A3 (ξ )a13 (y) + A4 (ξ )a14 (y)
2μA uxA = A1 (ξ )a21 (y) + A2 (ξ )a22 (y) + A3 (ξ )a23 (y) + A4 (ξ )a24 (y)
σ yA = A1 (ξ )a31 (y) + A2 (ξ )a32 (y) + A3 (ξ )a33 (y) + A4 (ξ )a34 (y)
τ xyA = A1 (ξ )a41 (y) + A2 (ξ )a42 (y) + A3 (ξ )a43 (y) + A4 (ξ )a44 (y)
σ wA = A1 (ξ )a51 (y) + A2 (ξ )a52 (y) + A3 (ξ )a53 (y) + A4 (ξ )a54 (y)
σ xA = A1 (ξ )a61 (y) + A2 (ξ )a62 (y) + A3 (ξ )a63 (y) + A4 (ξ )a64 (y).
(12)
Strip/half-plane B:
2μB uyB = B1 (ξ )b11 (y) + B3 (ξ )b13 (y)
2μB uxB = B1 (ξ )b21 (y) + B3 (ξ )b23 (y)
σ yB = B1 (ξ )b31 (y) + B3 (ξ )b33 (y)
τ xyB = B1 (ξ )b41 (y) + B3 (ξ )b43 (y)
σ wB = B1 (ξ )b51 (y) + B3 (ξ )b53 (y)
σ xB = B1 (ξ )b61 (y) + B3 (ξ )b63 (y).
(13)
Axisymmetric fretting analysis in coated cylinder
307
The corresponding expressions for aij and bij for strip and half-plane are given in the
Appendix A.2 and A.3. The boundary conditions in the Fourier space are:
Outer boundary: σ yA = P ; τ xyA = Q.
Interface: σ yA − σ yB = 0; τ xyA − τ xyB = 0; uyA − uyB = 0; uxA − uxB = 0.
3. Results and discussion
An infinitely long coated cylinder with coating A and core B is considered for analysis
with R1 and R2 being the outer and interface radii. The width of pressure band is taken as
a = 0·15R1 . The width of the slip region is taken equal to the coating thickness c = a/2.
The interface radius R2 = R1 − c. The coating thickness is generally of the order ranging
from few microns to a fraction of mm. In this analysis the coating thickness is equal to half
of the semi contact length. Both hard and soft coatings are considered. Four different elastic
mismatch ratios considered are μA /μB = 0·5, 1, 2, 5 with the Poisson’s ratio (ν = 0·25)
being same for both coating and substrate. Suffixes A and B are used to represent coating and
substrate, respectively. The cases of μA = 2μB , 5μB correspond to the case of hard coating
and μA = μ2B corresponds to the case of soft coating. The case of μA = μB corresponds to
a homogeneous cylinder which is considered as the standard with which other cases can be
compared. The coefficients of friction φ are 0, 0·3, 0·7 and ∞ (pure shear). The stresses
obtained from cylinder are compared with the stresses obtained from 2D models for particular
values of μ and φ. Different values of the ratios a/R1 and c/a are considered for illustrating
the results.
The matrix equation 7 is solved numerically and the arbitrary constants A1 to A4 , B1 and
B3 are obtained. The stress values have been evaluated by substituting the values obtained
for the arbitrary constants and by taking the corresponding inverse transforms of equations 3
and 4. The actual values of Bessel functions are used for calculating aij and bij . The range
of ξ is taken from 1 × 10−20 to 350 for implementing numerical calculations. Beyond 350
asymptotic expressions are used up to 1050. For evaluating the integrals of inverse transforms,
trapezoidal rule is used with the interval of ξ taken to be unity.
Similarly for the cases of strip and half-plane inversion of equations 12 to 13 gives the
stresses and displacements. For numerical inversion actual values of hyperbolic and exponential functions are used for ξ = 1 × 10−20 to 700 with the interval for numerical integration
by trapezoidal rule taken as unity.
We present the results in the vicinity of the edge of the loading zone in a rectangular
window 2a × a. It is a common practice to normalise stress with respect to the applied contact
pressure. In the case of coated components, however, it is also advantageous to examine
strains. In fretting contact problems the stress gradients near the contact are generally high.
Owing to this feature, crack initiation appears to depend on the state of stress developed in a
characteristic volume of material rather than a single material point on the surface, (Nowell
& Araujo 2003). Hence contour plots will reveal the areas over which high stress gradients
are present which can be used for multi-axial fatigue modelling (Papadopoulos et al 1997;
Goncalves et al 2005) to predict crack initiation. The contours of positive and negative values
are shown in red and blue colours. The contours corresponding to zero and near zero values
are shown in black. The green solid line shows the interface between the coating and the
substrate whereas the dotted green line in homogeneous solid shows the position of interface
which is present in the coated case. The contours of different stresses and strains normalised
with the applied pressure are presented in the range of −3 to 3 with 0·1 interval.
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M Ramesh, Satish V Kailas and K R Y Simha
Depending on the elastic mismatch between the coating and the substrate, the stresses in
the substrate need to be rescaled to be seen in the contour plots. High stresses are induced in
hard coatings while soft coatings are stressed little. It is therefore convenient to rescale the
stresses in the substrate by the ratio of moduli for stress components that are discontinuous
across the coating like σz and σθ (see Appendix B for scaling factors). The other two stresses
are not rescaled. Similarly, the strain r and τoct are rescaled. It should be noted that rescaling
reveals more detailed information.
3.1 Effect of elastic mismatch
The coating applied to the surface can be either soft or hard compared to the substrate. They
are found to respond to the fretting tractions differently. Soft coatings are generally thick and
hence if the interface is not sufficiently strong delamination can occur (Xu et al 2007). In
contrast, hard coatings are generally thin and may fail in a brittle manner when subjected to
fatigue loading (Golden et al 2007). Thus the mechanism of failure of soft and hard coatings
are different and hence it is important to study the effect of elastic mismatch. In this analysis
the Poisson’s ratio of the substrate and coating are assumed to be same and only the effect of
different shear moduli of coating and substrate is discussed.
Figure 5 shows contours of σr in the 2a × a window with its center coinciding under the
end of the pressure band along the interface. The coefficient of friction is taken as φ = 0·7
and four different elastic mismatch as mentioned earlier are considered. In general, coefficient
of friction is generally found to be high in fretting fatigue contact and hence a high value
of 0·7 is chosen (Dobromirsky & Smith 1986). The figure shows that for different elastic
mismatches the contours of σr look similar with only minor changes. Similar result is shown
for the τrz in figure 6. Also it can be seen that these stresses are continuous along the interface
as demanded by the boundary condition.
Figure 5. Contours of σr in cylinder (φ = 0·7, a = 0·15R1 and c = 0·5a).
Axisymmetric fretting analysis in coated cylinder
309
Figure 6. Contours of τrz in cylinder (φ = 0·7, a = 0·15R1 and c = 0·5a).
However for the case of σz , the elastic mismatch has a larger effect as seen from figure 7. It
can be shown that as the coating becomes much harder than the substrate, σz becomes tensile in
a region nearer to the interface. This shows the bending of the coating due to normal pressure
as the core becomes comparatively softer. For the case of σθ (figure 8) the stress component
remains compressive even with increasing elastic mismatch. It should be remembered that
stresses in the substrate are rescaled with the ratio of elastic mismatch for the cases of σz and
σθ . These stresses are discontinuous along the interface as expected.
Figure 7. Contours of σz in cylinder (φ = 0·7, a = 0·15R1 and c = 0·5a).
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M Ramesh, Satish V Kailas and K R Y Simha
Figure 8. Contours of σθ in cylinder (φ = 0·7, a = 0·15R1 and c = 0·5a).
The contours of τoct which controls plasticity and potential sites for crack initiation are
shown in figure 9. The figure shows that contours look similar for homogeneous and soft
coating case. However, the contours change shape significantly for the case of hard coating.
The contours in the core are plotted after rescaling with the corresponding modulus ratio.
Figure 10 shows the contours of dilatation where we see that it remains compressive for all
the four cases of elastic mismatch with the magnitudes increasing with increasing coating
Figure 9. Contours of τoct in cylinder (φ = 0·7, a = 0·15R1 and c = 0·5a).
Axisymmetric fretting analysis in coated cylinder
311
Figure 10. Contours of in cylinder (φ = 0·7, a = 0·15R1 and c = 0·5a).
hardness. The contours of strain components r and z are shown in figures 11 and 12. The
r in the core cylinder is rescaled with ratio of elastic shear modulus and z is not rescaled
since it is continuous at the interface. The r in soft coating is higher than homogeneous case
and it decreases with increasing μA elastic shear modulus of the coating. This shows that soft
coating is more compliant and would take any plastic deformation due to contact load without
deteriorating the substrate. But hard coatings are less compliant and more brittle which may
lead to failure of coating especially when they are subjected to fatigue loading.
Figure 11. Contours of r in cylinder (φ = 0·7, a = 0·15R1 and c = 0·5a).
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M Ramesh, Satish V Kailas and K R Y Simha
Figure 12. Contours of z in cylinder (φ = 0·7, a = 0·15R1 and c = 0·5a).
Again the value of z is highest for hard coating. This could be attributed to the bending of
the coating in presence of comparatively soft substrate. In the above discussion the hardness
is correlated with the elastic shear modulus (μ) since they are generally proportional.
3.2 Effect of coefficient of friction
The best way to reduce the severity of fretting is to reduce the coefficient of friction φ. The
coatings are generally expected to reduce the coefficient of friction. The hardness of the
Figure 13. Contours of σz in cylinder (μA = 0·5μB , a = 0·15R1 and c = 0·5a).
Axisymmetric fretting analysis in coated cylinder
313
Figure 14. Contours of τoct in cylinder (μA = 0·5μB , a = 0·15R1 and c = 0·5a).
coating also plays a role. Hence the effect of friction is studied here with reference to hard
and soft coating separately. Fretting problem always lies between the two extremes of only
pressure (figure 13a) and only shear loading (figure 13d) cases. Hence two different cases of
coefficient of friction (φ = 0·3, 0·7) are studied along with the cases of only pressure (φ = 0)
and only shear (without any pressure applied on the surface). The high friction coefficient can
be reduced by using solid lubricants. Hence a lower value of 0·3 is also used in the analysis.
Figure 15. Contours of σ1 in cylinder (μA = 0·5μB , a = 0·15R1 and c = 0·5a).
314
M Ramesh, Satish V Kailas and K R Y Simha
Figure 16. Contours of σz in cylinder (μA = 2μB , a = 0·15R1 and c = 0·5a).
Figure 13 shows contours of σz for the case of soft coating μA = 0·5μB . This shows as
the coefficient of friction is increased the σz becomes tensile starting from the trailing edge
of the shear profile and hence is important.
For soft coating the contours of τoct and σ1 are shown in figures 14 and 15. It can be seen
that as φ increases the stresses become more concentrated at the surface and σ1 becomes
tensile similar to σz .
Similar results are shown in figures 16, 17 and 18 for the case of hard coating (μA = 2μB ).
This shows that as the coefficient of friction increases the magnitude of shear loading leading
to further concentration of the stresses in the coating.
Figure 17. Contours of τoct in cylinder (μA = 2μB , a = 0·15R1 and c = 0·5a).
Axisymmetric fretting analysis in coated cylinder
315
Figure 18. Contours of σ1 in cylinder (μA = 2μB , a = 0·15R1 and c = 0·5a).
3.3 Effect of geometry
Fellows et al (1995) discussed the effect of finite thickness of the body on the calculation
of the contact stresses in the context of fretting. The results presented here provide further
insight regarding the tensile stress zones to quantify the severity of fretting. The geometry of
component can be cylindrical as in the case of shrink-fit shafts and sleeves rather than strip and
half-plane. Hence a comparison among the three geometries is attempted here to explore the
geometry-dependent effect on the state of stress near the surface including the interface. The
results of three geometries are compared for two different elastic mismatch and two different
values of coefficient of friction. Contours of τoct obtained from cylinder are compared with
the contours obtained from strip and half-plane (2D plane strain models). Figures 19 and 20
correspond to the cases of soft coating with coefficient of friction φ = 0·3 and 0·7 respectively.
The figures show that as the coefficient of friction is increased the contours in all the three
geometries become more similar. This is due to the fact that for the case of shear loading the
stress is concentrated at the surface and hence the actual geometry of the component becomes
less important.
Figures 21 and 22 correspond to the case of hard coating with φ = 0·3 and 0·7 respectively.
Here also we see that as the coefficient of friction increases τoct is more concentrated at the
surface.
3.4 Effect of a/R1 ratio
Four different values chosen for a/R1 ratio are 0·05, 0·1, 0·3 and 0·45 with c = a/2 and
R2 = 0·925R1 . Since the difference between the cylinder, strip and half-plane depends on the
a/R1 ratio it is important to study the effect of a/R1 . Smaller the a/R1 ratio better will be the
match expected between the strip, cylinder and half-plane. This can be seen from figures 23
and 24. The results are presented in a window of size 0·3R1 × 0·15R1 . This is because contact
width is changing with different values of a/R1 ratio.
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M Ramesh, Satish V Kailas and K R Y Simha
Figure 19. Contours of τoct (soft coating, low friction with
a = 0·15R1 and c = 0·5a).
3.5 Effect of c/a ratio
The actual nature of contact problem in fretting is generally a mixed boundary problem. The
existence of partial slip conditions complicates this situation further. Hence a non-linear way
of obtaining the contact parameters a and c is required. However in this work reasonable
values of c/a ratio (0·2, 0·4, 0·6 and 0·8) are considered individually for simplification and
the effect of the ratio is studied (figures 25 and 26).
Figure 20. Contours of τoct (soft coating, high friction with
a = 0·15R1 and c = 0·5a).
Axisymmetric fretting analysis in coated cylinder
317
Figure 21. Contours of τoct (hard coating, low friction with
a = 0·15R1 and c = 0·5a).
3.6 Sign reversal in shear stress
Fretting tractions may induce subsurface shear stresses of opposite sign. The distance of
separation being small which leads to change of sign of τrz as one moves from the surface to
the depth. This is true for different cases of elastic mismatch (figure 27). This phenomenon is
reminiscent of flow reversal in tubes. In this figure it should be noted that window is expanded
to cover the entire cylinder radius. Hence for numerical inversion actual values of Bessel
functions are used for ξ = 0 to 350 only and no asymptotic expressions are used.
Figure 22. Contours of τoct (hard coating, high friction with
a = 0·15R1 and c = 0·5a).
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M Ramesh, Satish V Kailas and K R Y Simha
Figure 23. Effect of a/R1 (c = 0·5a, μA = 2μB and φ = 0·7).
3.7 Interface delamination
It is important to study the tractions along the interface. The ratio of τrz /σr exceeding a critical
value forms the basis of Mohr–Coulomb theory of failure for brittle materials similar to the
Figure 24. Effect of a/R1 in strip (c = 0·5a, μA = 2μB and φ = 0·7).
Axisymmetric fretting analysis in coated cylinder
319
Figure 25. Effect of c/a (a = 0·15R1 , μA = 2μB and φ = 0·7).
coefficient of friction. A plot of τrz /σr ratio along the interface is shown in figure 28 for both
hard and soft coating corresponding to both high and low coefficients of friction. The ratio is
plotted for three different geometries using three different colours as shown in the legend. It
is interesting to note that τ/σ ratio reaches value well over 10φ as we move away from the
loading zone.
Figure 26. Effect of c/a in strip (a = 0·15R1 , μA = 2μB and φ = 0·7).
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M Ramesh, Satish V Kailas and K R Y Simha
Figure 27. Shear stress reversal (p0 = 0, a = 0·15R1 and c = 0·4a).
4. Conclusion
The effect of varying coating parameters like elastic mismatch between the coating and the
substrate, coefficient of friction and the loading geometry on the stresses is discussed in this
paper. The main results are:
• Elastic mismatch is important as σz becomes tensile due to bending effect.
• Increase in coefficient of friction leads to concentrated stress in the near surface region.
• The interfacial stress ratio τ/σ is increasing uniformly away from the loading zone
reaching values well over 10φ.
• The stresses roughly scale with the elastic shear modulus indicating the continuity of
strains across the interface in thin coatings.
Figure 28. Ratio of τrz /σr along interface.
Axisymmetric fretting analysis in coated cylinder
321
Based on the results obtained in this paper, we can conclude that coatings of thickness up to
10% of the cylinder radius may behave like thin shells. Therefore, it is possible to apply shell
theory for additional insight particularly concerning bending stresses. It should be noted that
shell elements used in FEM provide only an approximate model.
Nomenclature
2a
2c
R1
R2
p0
q0
P, Q
φ
μ
ν
E
A, B
r, θ, z
x, y, w
σr , σθ , σz
τrz
r , θ z
γrz
ur , uz
τoct
ξ
η
∇2
I0 , I1
K0 , K1
Contact width
Stick zone width
Radius of cylinder with coating
Radius of cylinder without coating
Applied pressure
Shear developed due to coefficient of friction
Pressure and shear tractions along the contact
Coefficient of friction
Shear modulus
Poisson’s ratio
Average of elastic moduli of coating and substrate
Represents coating and substrate
Cylindrical coordinates
Cartesian coordinates
Love/Airy stress function
Radial, tangential and axial stresses
Shear stress
Radial, tangential and axial strains
Shear strain
Radial and axial displacements
Dilatation
Octahedral shear stress
Variable of Fourier transform
Represents z in cylindrical coordinates
and x in Cartesian coordinates
Laplace’s operator
Modified Bessel functions of first kind
Modified Bessel functions of second kind
Appendix A. Expressions for aij and bij
Appendix A.1 Cylinder
a11 = −I1 (ξ r)/I0 (ξ R1 )/ξ
(14)
a12 = K1 (ξ r)/K0 (ξ R1 )/ξ
(15)
a13 = −ξ rI0 (ξ r)/I0 (ξ R1 )/ξ
(16)
a14 = ξ rK0 (ξ r)/K0 (ξ R1 )/ξ
(17)
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M Ramesh, Satish V Kailas and K R Y Simha
a21 = I0 (ξ r)/I0 (ξ R1 )/ξ
(18)
a22 = K0 (ξ r)/K0 (ξ R1 )/ξ
(19)
a23 = [4(1 − ν)I0 (ξ r) + ξ rI1 (ξ r)]/I0 (ξ R1 )/ξ
(20)
a24 = [−4(1 − ν)K0 (ξ r) + ξ rK1 (ξ r)]/K0 (ξ R1 )/ξ
(21)
a31
a32
I1 (ξ r)
/I0 (ξ R1 )
= −I0 (ξ r) +
ξr
K1 (ξ r)
/K0 (ξ R1 )
= −K0 (ξ r) −
ξr
(22)
(23)
a33 = [−(1 − 2ν)I0 (ξ r) − ξr I1 (ξ r)]/I0 (ξ R1 )
(24)
a34 = [(1 − 2ν)K0 (ξ r) − ξr K1 (ξ r)]/K0 (ξ R1 )
(25)
a41 = I1 (ξ r)/I0 (ξ R1 )
(26)
a42 = −K1 (ξ r)/K0 (ξ R1 )
(27)
a43 = [2(1 − ν)I1 (ξ r) + ξ rI0 (ξ r)]/I0 (ξ R1 )
(28)
a44 = [2(1 − ν)K1 (ξ r) − ξ rK0 (ξ r)]/K0 (ξ R1 )
(29)
a51 =
−I1 (ξ r)
/I0 (ξ R1 )
ξr
(30)
a52 =
K1 (ξ r)
/K0 (ξ R1 )
ξr
(31)
a53 = −(1 − 2ν)I0 (ξr )/I0 (ξ R1 )
(32)
a54 = (1 − 2ν)K0 (ξr )/K0 (ξ R1 )
(33)
a61 = I0 (ξ r)/I0 (ξ R1 )
(34)
a62 = K0 (ξ r)/K0 (ξ R1 )
(35)
a63 = [2(2 − ν)I0 (ξ r) + ξ rI1 (ξ r)]/I0 (ξ R1 )
(36)
a64 = [−2(2 − ν)K0 (ξ r) + ξ rK1 (ξ r)]/K0 (ξ R1 )
(37)
b11 = −I1 (ξ r)/I0 (ξ R1 )/ξ
(38)
b13 = −ξ rI0 (ξ r)/I0 (ξ R1 )/ξ
(39)
Axisymmetric fretting analysis in coated cylinder
323
b21 = I0 (ξ r)/I0 (ξ R1 )/ξ
(40)
b23 = [4(1 − ν)I0 (ξ r) + ξ rI1 (ξ r)] /I0 (ξ R1 )/ξ
(41)
I1 (ξ r)
b31 = −I0 (ξ r) +
/I0 (ξ R1 )
ξr
(42)
b33 = [−(1 − 2ν)I0 (ξ r) − ξ rI1 (ξ r)] /I0 (ξ R1 )
(43)
b41 = I1 (ξ r)/I0 (ξ R1 )
(44)
b43 = [2(1 − ν)I1 (ξr ) + ξ rI0 (ξ r)] /I0 (ξ R1 )
(45)
−I1 (ξ r)
/I0 (ξ R1 )
ξr
(46)
b53 = −(1 − 2ν)I0 (ξ r)/I0 (ξ R1 )
(47)
b61 = I0 (ξ r)/I0 (ξ R1 )
(48)
b63 = [2(2 − ν)I0 (ξ r) + ξ rI1 (ξ r)] /I0 (ξ R1 )
(49)
b51 =
Appendix A.2 Strip
a11 = e−ξ(y−Y1 ) /ξ
(50)
a12 = −eξ(y−Y1 ) /ξ
(51)
a13 = [ξy + (1 − 2ν)]e−ξ(y−Y1 ) /ξ
(52)
a14 = [−ξy + (1 − 2ν)]eξ(y−Y1 ) /ξ
(53)
a21 = e−ξ(y−Y1 ) /ξ
(54)
a22 = eξ(y−Y1 ) /ξ
(55)
a23 = [ξy − 2(1 − ν)]e−ξ(y−Y1 ) /ξ
(56)
a24 = [ξy + 2(1 − ν)]eξ(y−Y1 ) /ξ
(57)
a31 = −e−ξ(y−Y1 )
(58)
a32 = −eξ(y−Y1 )
(59)
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M Ramesh, Satish V Kailas and K R Y Simha
a33 = −ξye−ξ(y−Y1 )
(60)
a34 = −ξyeξ(y−Y1 )
(61)
a41 = −e−ξ(y−Y1 )
(62)
a42 = eξ(y−Y1 )
(63)
a43 = (1 − ξy)e−ξ(y−Y1 )
(64)
a44 = (1 + ξy)eξ(y−Y1 )
(65)
a51 = 0
(66)
a52 = 0
(67)
a53 = −2νe−ξ(y−Y1 )
(68)
a54 = 2νeξ(y−Y1 )
(69)
a61 = e−ξ(y−Y1 )
(70)
a62 = eξ(y−Y1 )
(71)
a63 = (ξy − 2)e−ξ(y−Y1 )
(72)
a64 = (ξy + 2)eξ(y−Y1 )
(73)
b11 = − sinh(ξy)/eξ Y1 /ξ
(74)
b13 = −[ξy cosh(ξy) − (1 − 2ν) sinh(ξy)]/eξ Y1 /ξ
(75)
b21 = cosh(ξy)/eξ Y1 /ξ
(76)
b23 = [ξy sinh(ξy) + 2(1 − ν) cosh(ξy)]/eξ Y1 /ξ
(77)
b31 = − cosh(ξy)/eξ Y1
(78)
b33 = −ξy sinh(ξy)/eξ Y1
(79)
b41 = sinh(ξy)/eξ Y1
(80)
b43 = [ξy cosh(ξy) + sinh(ξy)]/eξ Y1
(81)
Axisymmetric fretting analysis in coated cylinder
325
b51 = 0
(82)
b53 = 2ν cosh(ξy)/eξ Y1
(83)
b61 = cosh(ξy)/eξ Y1
(84)
b63 = [ξy sinh(ξy) + 2 cosh(ξy)]/eξ Y1
(85)
Appendix A.3 Half-plane
a11 = e−ξy /ξ
(86)
a12 = −eξy /ξ
(87)
a13 = (1 − 2ν + ξy)e−ξy /ξ
(88)
a14 = (1 − 2ν − ξy)eξy /ξ
(89)
a21 = e−ξy /ξ
(90)
a22 = eξy /ξ
(91)
a23 = [ξy − 2(1 − ν)]e−ξy /ξ
(92)
a24 = [ξy + 2(1 − ν)]eξy /ξ
(93)
a31 = −e−ξy
(94)
a32 = −eξy
(95)
a33 = −ξye−ξy
(96)
a34 = −ξyeξy
(97)
a41 = −e−ξy
(98)
a42 = eξy
(99)
a43 = (1 − ξy)e−ξy
(100)
a44 = (1 + ξy)eξy
(101)
a51 = 0
(102)
a52 = 0
(103)
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M Ramesh, Satish V Kailas and K R Y Simha
a53 = −2νe−ξy
(104)
a54 = 2νeξy
(105)
a61 = e−ξy
(106)
a62 = eξy
(107)
a63 = (ξy − 2)e−ξy
(108)
a64 = (ξy + 2)eξy
(109)
b11 = e−ξy /ξ
(110)
b13 = (1 − 2ν + ξy)e−ξy /ξ
(111)
b21 = e−ξy /ξ
(112)
b23 = [ξy − 2(1 − ν)]e−ξy /ξ
(113)
b31 = −e−ξy
(114)
b33 = −ξye−ξy
(115)
b41 = −e−ξy
(116)
b43 = (1 − ξy)e−ξy
(117)
b51 = 0
(118)
b53 = −2νe−ξy
(119)
b61 = e−ξy
(120)
b63 = −(2 − ξy)e−ξy
(121)
Appendix B. Scaling factors
soft coating
coating
substrate
hard coating
stress
strain
stress
strain
μB /μA
1
1
μB /μA
1
μA /μB
μA /μB
1
Axisymmetric fretting analysis in coated cylinder
327
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